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Trigonometry and 3-Space 122 - TRIGONOMETRY REVIEW
1. Convert 140o to radians.
2. Convert
2
to degrees.
5
3. What is the smallest positive angle that is coterminal with 1600o?
4. Find the exact value of each of the following.
a. csc 225o
b. sec120o
c. cot
4
3
5. Simplify the following rational expressions and state any restrictions on x.
a.
x 2  3x
x 3
 2
3
2
x  2 x  3x x  3x  2
b.
x
2x  3
 2
2x  2 2x  6x  8
6. Find the exact value of each of the following.
a. sin
11
12
b. cos 75o
7. Without the aid of a calculator, find the exact value of each expression.
Simplify your final answer and rationalize any denominator.
a.
o
3sin 45
sin 90o  2 cos 30o
b.
sin
5
11
 cos 2
4
6
5sin

6
8. You are on a research submarine which is doing submerged training exercises
in the Pacific Ocean. While submerged, at time t = 0, you start porpoising
(alternating deeper and then shallower). At time t = 4 minutes you first reach
your deepest, d = –1000 m. At time t = 10 minutes, you next reach your
shallowest, d = –200 m. Assume that the submarine’s depth varies
sinusoidally with time.
a. Sketch a graph of this function through at least one full period. Clearly
label your axes, identify your scale, and mark all key points on your graph.
b. Write one equation that could represent this situation.
c. Your submarine can’t communicate with ships on the surface when it is
deeper than d = –300 m. At time t = 45 minutes, could your submarine
communicate?
9. Solve the following equations for  measured in degrees.
a. 1  2sin 2  
2
2
b.
sin(5  30o ) 
2
2
c. tan   tan 2   0
10. Solve the following equations for   [2 , 2 ] .
a. sin 2  
3
2
b. 2 cos 2   cos   1  0
c. 2sin  cos   2 cos   0
11. Prove the following identities. State any restrictions on x.
a.
1  sin2 x
 sin x  cos x
sin x  cos x
b.
sin x
1  cos x

1  cos x
sin x
12. Prove the following identities.
a. 2 cos x  2 tan 2 x cos x  2sec x
c.
sin 2 x cos 2 x

 sec x
sin x
cos x
b. sin x  cot x cos x  csc x
d. sec 2 x  tan 2 x 
cos x  sin x
cos x  sin x
13 i) State the period of each of the following functions.
ii) Sketch the graph of each function. Show at least two full periods.
a. 3( y  2)  cos 4( x  45o )
c.
1

( y  2)  csc( x  )
3
2

b. y  1  tan 3( x  )
4
1

d. 2( y  3)  cot ( x  )
2
3
14. Write a possible equation for each graph.
y
a.
4
2
x
-20
-15
-10
-5
0
5
10
15
20
-2
-4
-6
-8
y
b.
2
1
x













y
c.
6
5
4
3
2
1
x (degrees)
-270
-180
-90
0
-1
-2
90
180
270